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Many natural phenomena can be considered to be complex systems, and their study (complexity science) is highly interdisciplinary. Examples of complex systems include ant-hills, ants themselves, humaneconomies, nervous systems, cells and living things, including human beings.

Beyond the fact that these things are all networks of some kind, and that they are complex, it may appear that they have little in common, hence that the term "complex system" is vacuous. However, all complex systems are held to have behavioural and structural features in common, which at least to some degree unites them as phenomena. They are also united theoretically, because all these systems may, in principle, be modelled with varying degrees of success by a certain kind of mathematics. It is therefore possible to state clearly what it is that these systems are supposed to have in common with each other, in relatively formal terms.

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The term complex system formally refers to a system of many parts which are coupled in a nonlinear fashion. Natural complex systems are modelled using the mathematical techniques of dynamical systems, which include differential equations, difference equations and maps. Because they are nonlinear, complex systems are more than the sum of their parts because a linear system is subject to the principle of superposition, and hence is literally the sum of its parts, while a nonlinear system is not. Put another way: a linear relationship is simply one whose graph is a straight line, so a linear connection between two things is one in which change on one side of the connection induces proportional change in the other. A nonlinear connection means that change on one side is not proportional to change on the other. When there are many non-linearities in a system (many components), behaviour can be as unpredictable as it is interesting. Complex systems research studies such behaviour.

Most biological systems are complex systems in the sense outlined above, while traditionally, most humanly engineered systems are not. Complex systems research overlaps substantially with nonlinear dynamics research, but complex systems specifically consist of a large number of mutually interacting dynamical parts. Many research disciplines are becoming interested in this branch of mathematical analysis because the digital computer has made theoretical exploration of such systems possible. A little mathematical knowledge is required to see why that is so. See for example numerical integration.

Variables in complex systems may of course exhibit very complex, discoordinated behaviours, in which it is very hard to predict what an element will do over time. However, Complex Systems may also exhibit relatively simple (or, more formally, low dimensional, or coordinated) patterns of behaviour just as simpler, linear systems do. But, unlike linear systems (however complicated), complex non-linear systems are usually very flexible in terms of exhibiting qualitatively different behaviours at different times. In dynamical systems terminology, such a qualitative change is known as a bifurcation and non-linearity is required for a system to exhibit it. Part of complex systems research is to determine whether any simple rules exist which may be used to describe the low dimensional behaviour of complex systems. See for example coordination dynamics. Because of the presence of low dimensional behaviours and putative rules governing global behaviour, many refer to the properties of complex systems as emergent. It is important to realise though, that the use of this term is contentious within the complex systems research community, since in principle knowledge of the properties of the components and coupling between them is sufficient to determine all aspects of the system's behaviour. In practice though, this is often not possible (or desirable) and it is not wholly misleading to see complex systems research as the study of "emergent" properties.

The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neuroscience, meteorology, physics, computer science, artificial life, evolutionary computation, economics, heart cell synchronisation, immune systems, molecular biology, epilepsy and enquiries into the nature of living cells themselves. In these endeavours, scientists often seek simple non-linear coupling rules which lead to complex phenomena (rather than describe - see above), but this need not be the case. Human societies (and probably human brains) are complex systems in which neither the components nor the couplings are simple. Nevertheless, they exhibit many of the hallmarks of complex systems.

Traditionally, engineering has striven to keep its systems linear, because that makes them simpler to build and to predict. However, many physical systems (for example lasers) are inherently "complex systems" in terms of the definition above, and engineering practice must now include elements of complex systems research.

In practical terms, this means a small perturbation may cause a large effect (see butterfly effect), a proportional effect, or even no effect at all. In linear systems, effect is always directly proportional to cause. See nonlinearity.

Complex systems in nature are usually open systems — that is, they exist in a thermodynamic gradient and dissipate energy. In other words, complex systems are usually far from energetic equilibrium: but despite this flux, there may be pattern stability. See synergetics.

The history of a complex system may be important. Because complex systems are dynamical systems they change over time, and prior states may have an influence on present states. More formally, complex systems often exhibit hysteresis.

The components of a complex system may themselves be complex systems. For example, an economy is made up of organisations, which are made up of people, which are made up of cells - all of which are complex systems.

As well as coupling rules, the dynamic network of a complex system is important. small-world or scale-free networks which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human cortex for example, we see dense local connectivity and a few very long axon projections between regions inside the cortex and to other brain regions.

From Sync by Steven Strogatz: "Every decade or so, a grandiose theory comes along, bearing similar aspirations and often brandishing an ominous-sounding C-name. In the 1960 it was cybernetics. In the '70s it was catastrophe theory. Then came chaos theory in the '80s and complexity theory in the '90s."

Various informal descriptions of complex systems have been put forward, and these may give some insight into their properties. A special edition of Science about complex systemsScience Vol. 284. No. 5411 (1999).
highlighted several of these:

A complex system is a highly structured system, which shows structure with variations (Goldenfeld and Kadanoff)

A complex system is one whose evolution is very sensitive to initial conditions or to small perturbations, one in which the number of independent interacting components is large, or one in which there are multiple pathways by which the system can evolve (Whitesides and Ismagilov)

A complex system is one that by design or function or both is difficult to understand and verify (Weng, Bhalla and Iyengar)

A complex system is one in which there are multiple interactions between many different components (D. Rind)

Complex systems are systems in process that constantly evolve and unfold over time (W. Brian Arthur).